Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Partial Differential Equations Laplace Equation, Heat and Wave Equations
तरंग समीकरण के लिए कॉची समस्या पर विचार कीजिए,
\(\frac{\partial^2 u}{\partial t^2}-4 \frac{\partial^2 u}{\partial x^2}=0\) , -∞ < x <∞, t > 0,
\(u(x, 0)=\left\{\begin{array}{cc} e^{\left(-\frac{1}{x^2}\right)}, & x \neq 0, \\ 0, & x=0, \end{array}\right.\)
\(\frac{\partial u}{\partial t}(x, 0)=x e^{-x^2}\) , x ∈ ℝ
निम्नलिखित में से कौन सा सत्य है?
1
\(\displaystyle \lim _{t \rightarrow \infty} u(5, t)=1\)
2
\(\displaystyle \lim _{t \rightarrow \infty} u(5, t)=2\)
3
\(\displaystyle \lim _{t \rightarrow \infty} u(5, t)=\frac{1}{2}\)
4
\(\displaystyle \lim _{t \rightarrow \infty} u(5, t)=0\)